# Gravity between two unequal masses. Do both masses move?

I've been watching videos about gravity and I have a question
My understanding is that mass have gravity and gravity is a force which attract other object with mass. For example, I jump up and the Earth's gravity pulls me down.

1. So my question is, is it always the case that the smaller mass that move towards the bigger mass?

2. Does the bigger mass EVER move towards the smaller mass?

3. If two objects with same mass are left in a vacuum, they meet in the middle point of the distance, right?

4. so what if one of the object has little bit more mass? i would assume the bigger mass would still move towards the middle point (but bit shorter)

5. If the above is true, can we technically move the Earth by us (human population) jumping indefinitely?

Though, since Earth's mass is 5.972x10^24 and the mass for human population would be around 4.9x10^11 (assuming 70kg avg weight for 7 billion people), it would have a minimal effect but given that we would jump infinity, we can technically move it, I think?

• see as well this and this Nov 20, 2015 at 17:01
• If you don't take the landing push away force you may as well disregard the jump whatsoever. Gravitation is a completely different thing and has nothing to do with pulling or pushing and should be counted separately from jumping. Better formed question would be 'Gather or humans into one logical point with the mass'
– Zloj
Nov 20, 2015 at 21:58
• If the above is true, can we technically move the Earth by us(human population) jumping indefinitely? Obligatory fun read (don't forget to read the first two links for the actual physics involved) Nov 20, 2015 at 22:27
• If all 7 billion people stood on the same side of the earth and jumped, the earth would be pushed a tiny bit the other way. Then when we all fell back, the earth would fall back as well and we would all be back where we started. We would move the earth for a little while, but we couldn't move it farther by jumping again. Nov 21, 2015 at 5:19
• Related: physics.stackexchange.com/q/28519/2451 and links therein. Nov 24, 2015 at 12:11

In all cases, the two objects move towards one another. In fact they experience exactly the same gravitational force. However, because acceleration equals force over mass $$\mathbf{a} = \frac{\mathbf{F}}{m}$$

that equal forces causes the heavier object to accelerate much less than the lighter one. But technically, the Earth does move towards you very slightly when you jump. However, it first moves slightly away from you because in order to jump you have to push it. By the time you land it returns to its original position.

The two objects will meet at their centre of gravity. That is to say if, for example, one mass is twice as big as the other, the meeting point will be one quarter of the way from the heavy mass to the light one. In general it is the point where

$$m_1 * r_1 = m_2 * r_2$$

We can't move the Earth by jumping indefinitely because the push-away from the jump exactly cancels the pull-towards from gravity. There is no net motion. This is the same reason you can't move a boat by sitting inside and kicking the walls.

• I mean, in a perfect vacuum with a spherical cow you are correct. The earth does not move "away" when you push on it, some of it does but most of that energy is taken by compression which translates into heat. So technically you could move the earth by jumping.
– Sam
Nov 20, 2015 at 23:34
• @Sam, [Conservation of Momentum](scienceworld.wolfram.com/physics/ConservationofMomentum.html) says even if you break the forces up into a billion billion sub-atomic interactions, the total acceleration of the system is zero. Even if jumping only generates "heat" (which is just semi-random, low-level kinetic energy), there's still a net downward momentum transfer to the planet as the person jumps up, then an upward transfer as the person falls down, then another downward transfer as the person lands, which balances to zero. Nov 21, 2015 at 2:29
• @MichaelS, none of that invalidates what I said. Conservation of momentum does apply but you are thinking about it wrong.
– Sam
Nov 21, 2015 at 14:43
• Instead of saying "you're wrong", try explaining it. What part of pushing -> compression -> heat causes the momentum of the Earth/person system to change? Or if the momentum doesn't change, how are you moving the Earth? Nov 22, 2015 at 8:20
• @Sam I think MichaelS is right here. Your argument would seem to equally imply that one could move a wagon by sitting inside of it and kicking the walls. You can't, because the momentum (unlike the heat) isn't dissipated.
– AGML
Nov 30, 2015 at 19:39

There is a mutual attraction from gravity, and we generally only consider the smaller object here on earth because the earth is so massive, the acceleration of the earth is negligible. This is because $a = F/m$, and with equal $F$ between the two objects, the acceleration will scale as $a\propto 1/m$. For the earth, this leaves $a$ ridiculously small, but technically non-zero.

There are some good example cases of both objects moving toward each other, and one particular case is the Pluto-Charon system. In this case, Pluto is more massive that Charon, but both objects are in mutual orbit and constantly "falling" toward one another. This can be observed based on the fact that both objects orbit a point outside of either mass as seen below (publicly available image from wikipedia): Now, these objects have angular momentum, so they will never meet, but I think it is a good example of how a small mass affects a larger mass under the influence of gravity.

Does the bigger mass EVER move towards the smaller mass?

Yes.

$F = KMm/r^2$

$M*a_{M}=F$

$m*a_{m}=F$

As you see the smaller the mass the higher the acceleration and in consequence the higher the traveled distance in a given time t.

If the above is true, can we technically move the Earth by us(human population) jumping indefinitely?

No.

Each time we jump, assuming all of us are in the same place and jump in a synchronized way, the earth will go a bit in the opposite direction but it comes back and when we land again the earth will be in the same position where it was before we jumped.

For the full explanation you have to add, to the equations above, the conservation of momentum and solve the system.

$M*v_{M} = m*v_{m}$ (conservation of momentum for the moment we jump)

You can also use the already demonstrated fact that the center of mass, for a system of N masses, does not move in time if no external force exist. In the case of the earth - people system, there is no external force and no matter where we are and how we jump the center of mass of all people + earth will always stay in the same place.

• Can you explain further about earth bbeing back in the same position when we land? How so? Nov 20, 2015 at 18:09
• @ealeon: You provide a force on the earth to move away from you; due to Newton's Third Law, you would feel the same force & as a result, you would move up but ultimately you would be decelerated by gravity; same is in the case of Earth also.
– user36790
Nov 20, 2015 at 18:12
• @user36790 makes sense now. didnt think about the initial force we would put on to the Earth as we jump. thank you! kinda cool that it would cancel out Nov 20, 2015 at 18:39

yes, the earth will accelerate towards you , however the Earth's acceleration will be so small for all practical purposes that you usually do not consider it. Earth's acceleration is small because the mutual forces between you and the earth are the same, but the masses are different, so this results in different accelerations (remember: $F=ma$). Now if you could put all humans in on place and jump, will this accelerate earth's in a noticeable way?Assume there are 7 billion humans, each averaging $100~\text{kgm}\;,$ the total mass will be $7 \space 10^{11}~\text{kg}\; .$ The earth's acceleration will be proportional $GM_\text{humankind}/R_\text{earth }^2$ which is approximately $0.007~\text{m}/\text{s}^2$, which is small but still measurable.